A, B and C can do a job working alone in 6, 9 and 18 days respectively. They all work together for 1 day, then A and B quit. How many days C working alone will take to complete the remainder of the job?

Options

A. 9
B. 6
C. 12
D. 10

Answer

To solve the problem, we need to determine how much work A, B, and C can do together in one day, and then find out how long C will take to finish the remaining work after A and B quit. Let's break it down step by step.

### Step 1: Determine the work done by A, B, and C in one day.

- A can complete the job in 6 days. Therefore, in one day, A can do:
\[ \text{Work done by A in 1 day} = \frac{1}{6} \text{ of the job} \]

- B can complete the job in 9 days. Therefore, in one day, B can do:
\[ \text{Work done by B in 1 day} = \frac{1}{9} \text{ of the job} \]

- C can complete the job in 18 days. Therefore, in one day, C can do:
\[ \text{Work done by C in 1 day} = \frac{1}{18} \text{ of the job} \]

### Step 2: Calculate the total work done by A, B, and C together in one day.

Now, we add the work done by A, B, and C together:
\[ \text{Total work in 1 day} = \frac{1}{6} + \frac{1}{9} + \frac{1}{18} \]

To add these fractions, we need a common denominator. The least common multiple (LCM) of 6, 9, and 18 is 18.

- Convert each fraction:
\[ \frac{1}{6} = \frac{3}{18}, \quad \frac{1}{9} = \frac{2}{18}, \quad \frac{1}{18} = \frac{1}{18} \]

- Now add them:
\[ \text{Total work in 1 day} = \frac{3}{18} + \frac{2}{18} + \frac{1}{18} = \frac{6}{18} = \frac{1}{3} \text{ of the job} \]

### Step 3: Determine how much work is left after 1 day.

Since A, B, and C worked together for 1 day and completed \(\frac{1}{3}\) of the job, the remaining work is:
\[ \text{Remaining work} = 1 - \frac{1}{3} = \frac{2}{3} \text{ of the job} \]

### Step 4: Calculate how long C will take to finish the remaining work.

C can complete \(\frac{1}{18}\) of the job in one day. To find out how many days C will take to complete \(\frac{2}{3}\) of the job, we set up the equation:
\[ \text{Days for C} = \frac{\text{Remaining work}}{\text{Work done by C in 1 day}} = \frac{\frac{2}{3}}{\frac{1}{18}} \]

### Step 5: Simplify the calculation.

To simplify:
\[ \text{Days for C} = \frac{2}{3} \times 18 = 2 \times 6 = 12 \text{ days} \]

### Final Answer:
C will take **12 days** to complete the remainder of the job.

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